So it should be all right for a few years if I just stick it in the fridge where I keep all my film?

that's it. just put it there. 2 or 3 years over the date are usually no problem if it has been stored properly. but if it's older than that it changes and you have to recalibrate your proceedings. and then it becomes useless because it dries out.

this is for sheet film packs. integral film (instax) is more sensible to age and integral film with batteries (polaroid and impossible) is worse.